Algorithms for nonlinear semi-definite programming

Obor
Řídicí technika a robotika

Outline

The candidate should first survey polynomial matrix inequality (PMI) formulations of standard linear and nonlinear systems control problems (e.g. robustness analysis, fixed-order controller design) using both the polynomial approach to systems control (Routh-Hurwitz and Hermite criterion, polynomial Diophantine equations, spectral factorization) and results of real algebraic geometry (Sturm sequences, Descartes rule of signs, polynomial positivity conditions). The various resulting formulations should then be compared and studied in deep detail, with a particular focus on geometric characteristics (measures and estimates of conditioning, design of simple preconditioners). The ultimate objective would be to assess the impact of this numerical analysis on the behavior of computer-aided control system design numerical algorithms (based on linear algebra, convex and nonconvex optimization). Throughout the study, the candidate is expected to develop and implement a collection of illustrative algorithms (in Matlab, Scilab, Maple or Mathematica) and apply them on difficult (sometimes open) control problems.

Requirements

The candidate must have a good background in systems control, linear algebra, numerical analysis, and applied mathematics in general. Programming skills (Matlab, Scilab, Maple, Mathematica) are appreciated. Some knowledge of algebraic geometry and convex optimization is welcome but not required. Most of the PhD work can be carried out at the Czech Technical University in Prague, but several short-stays (between two and four weeks) are expected at LAAS-CNRS in Toulouse, France, possibly in the scope of a French-Czech co-tutelle agreement.